Find a light-like geodesic in the sumbanifold $\R\times S^{n-1}\colon=\{(x_0,x_1,\ldots,x_n)\in\R_1^{n+1}: x_1^2+\cdots+x_n^2=1\}$ of $\R_1^{n+1}$.
Suppose $c(s)=(c_0(s),c_1(s),\ldots,c_n(s))$ is a geodesic; as the normal unit vectorfield to $M\colon=\R\times S^{n-1}$ is
$$
\forall x\in M:\quad
N_x=\sum_{j=1}^n x_j\,E_x^j
$$
the covariant derivative of
$$
c^\prime(s)=\sum_{j=0}^n c_j^\prime(s)\,E_{c(s)}^j
$$
is given by
$$
\bnabla c^\prime(s)
=c^\dprime(s)-\la c^\dprime(s),N\ra N_{c(s)}
$$
and thus the geodesic equation comes down to the set of equations $c_0^\dprime=0$ and
$$
\forall j=1,\ldots,n:\quad
0=c_j^\dprime-\sum_{k=1}^n c_k^\dprime c_kc_j~.
$$
For $j=1,\ldots,n$ these equations say that $k(s)\colon=(c_1(s),\ldots,c_n(s))$ is a geodesic in $S^{n-1}$. Hence we may take $c_0(s)=s$, $c_1(s)=\cos(s)$, $c_2(s)=\sin(s)$ and for all $j > 2$: $c_j(s)=0$.